Abstract In d - Scattered Set we are given an (edge-weighted) graph and are asked to select at least k vertices, so that the distance between any pair is at least d , thus generalizing Independent Set . We provide upper and lower bounds on the complexity of this problem with respect to various standard graph parameters. In particular, we show the following: • For any d ≥ 2 , an O ∗ ( d tw ) -time algorithm, where tw is the treewidth of the input graph and a tight SETH-based lower bound matching this algorithm’s performance. These generalize known results for Independent Set . • d - Scattered Set is W[1]-hard parameterized by vertex cover (for edge-weighted graphs), or feedback vertex set (for unweighted graphs), even if k is an additional parameter. • A single-exponential algorithm parameterized by vertex cover for unweighted graphs, complementing the above-mentioned hardness. • A 2 O ( td 2 ) -time algorithm parameterized by tree-depth ( td ), as well as a matching ETH-based lower bound, both for unweighted graphs. We complement these mostly negative results by providing an FPT approximation scheme parameterized by treewidth. In particular, we give an algorithm which, for any error parameter ϵ > 0 , runs in time O ∗ ( ( tw ∕ ϵ ) O ( tw ) ) and returns a d ∕ ( 1 + ϵ ) -scattered set of size k , if a d -scattered set of the same size exists.

中文翻译：

---

结构参数化的 d-scattered 集

摘要 在 d-Scattered Set 中，我们给定了一个（边加权）图，并要求选择至少 k 个顶点，以便任何对之间的距离至少为 d ，从而推广独立集。我们针对各种标准图形参数提供了该问题复杂性的上限和下限。特别地，我们展示了以下内容： • 对于任何 d ≥ 2 ，一个 O ∗ ( d tw ) - 时间算法，其中 tw 是输入图的树宽和一个与该算法性能匹配的基于 SETH 的紧密下界。这些概括了独立集的已知结果。• d - Scattered Set 是 W[1]-hard 参数化的顶点覆盖（对于边加权图）或反馈顶点集（对于未加权图），即使 k 是附加参数。• 由未加权图的顶点覆盖参数化的单指数算法，补充上述硬度。• 2 O ( td 2 ) 时间算法，由树深度 ( td ) 参数化，以及匹配的基于 ETH 的下界，均用于未加权图。我们通过提供由树宽参数化的 FPT 近似方案来补充这些主要是负面的结果。特别地，我们给出了一个算法，对于任何误差参数 ϵ > 0 ，在时间 O ∗ ( ( tw ∕ ϵ ) O ( tw ) ) 中运行并返回 ad ∕ ( 1 + ϵ ) - 大小为 k 的分散集，如果存在相同大小的广告分散集。